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E-raamat: Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Perturbations

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Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of PerturbationsSuurem pilt
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Because of the correspondences existing among all levels of reality, truths pertaining to a lower level can be considered as symbols of truths at a higher level and can therefore be the "foundation" or support leading by analogy to a knowledge of the latter. This confers to every science a superior or "elevating" meaning, far deeper than its own original one. - R. GUENON, The Crisis of Modern World Having been interested in the Kepler Problem for a long time, I have al­ ways found it astonishing that no book has been written yet that would address all aspects of the problem. Besides hundreds of articles, at least three books (to my knowledge) have indeed been published al­ ready on the subject, namely Englefield (1972), Stiefel & Scheifele (1971) and Guillemin & Sternberg (1990). Each of these three books deals only with one or another aspect of the problem, though. For example, En­ glefield (1972) treats only the quantum aspects, and that in a local way. Similarly, Stiefel & Scheifele (1971) only considers the linearization of the equations of motion with application to the perturbations of celes­ tial mechanics. Finally, Guillemin & Sternberg (1990) is devoted to the group theoretical and geometrical structure.

Arvustused

"This is an interesting book, which well organizes the group-geometric aspects of the Kepler problem on which a great number of articles have been published along with the advance of symmetry theory. . . . a nice reference not only for graduate students but also for scientists who are interested in dynamical systems with symmetry." --MathSciNet

Muu info

Springer Book Archives
Preface vii
List of Figures
xv
1 Introductory Survey
1(16)
1.1 Part I - Elementary Theory
2(3)
1.1.1 Basic Facts
2(1)
1.1.2 Separation of Variables and Action-Angle Variables
3(1)
1.1.3 Quantization of the Kepler Problem
4(1)
1.1.4 Regularization and Symmetry
5(1)
1.2 Part II - Group-Geometric Theory
5(6)
1.2.1 Conformal Regularization
5(2)
1.2.2 Spinorial Regularization
7(1)
1.2.3 Return to Separation of Variables
8(1)
1.2.4 Geometric Quantization
9(1)
1.2.5 Kepler Problem with a Magnetic Monopole
10(1)
1.3 Part III - Perturbation Theory
11(3)
1.3.1 General Perturbation Theory
11(1)
1.3.2 Perturbations of the Kepler Problem
12(1)
1.3.3 Perturbations with Axial Symmetry
13(1)
1.4 Part IV - Appendices
14(3)
1.4.1 Differential Geometry
14(1)
1.4.2 Lie Groups and Lie Algebras
15(1)
1.4.3 Lagrangian Dynamics
15(1)
1.4.4 Hamiltonian Dynamics
16(1)
I Elementary Theory
17(92)
2 Basic Facts
18(18)
2.1 Conies
18(4)
2.2 Properties of the Keplerian Motion
22(5)
2.2.1 Energy H < 0
23(2)
2.2.2 Energy H > 0
25(1)
2.2.3 Energy H = 0
26(1)
2.3 The Three Anomalies
27(2)
2.3.1 Energy H < 0
27(1)
2.3.2 Energy H > 0
27(1)
2.3.3 Energy H = 0
28(1)
2.4 The Elements of the Orbit for H < 0
29(3)
2.5 The Repulsive Potential
32(4)
Appendix
2.A The Kepler Equation
34(2)
3 Separation of Variables and Action--Angle Coordinates
36(25)
3.1 Separation of Variables
37(7)
3.1.1 Spherical Coordinates
37(1)
3.1.2 Parabolic Coordinates
38(2)
3.1.3 Elliptic Coordinates
40(2)
3.1.4 Spheroconical Coordinates
42(2)
3.2 Action--Angle Variables
44(17)
3.2.1 Delaunay and Poincare Variables
44(11)
3.2.2 Pauli Variables
55(4)
3.2.3 Monodromy
59(2)
4 Quantization of the Kepler Problem
61(35)
4.1 The Schrodinger Quantization
61(17)
4.1.1 Spherical Coordinates
66(5)
4.1.2 Parabolic Coordinates
71(2)
4.1.3 Elliptic Coordinates
73(2)
4.1.4 Spheroconical Coordinates
75(3)
4.2 Pauli Quantization
78(3)
4.2.1 Canonical Quantization
78(2)
4.2.2 Pauli Quantization
80(1)
4.3 Fock Quantization
81(15)
Appendix
4.A Mathematical Review
87(1)
4.A.1 Second Order Linear Differential Equations
87(2)
4.A.2 Laplacian on the Sphere and Homogeneous Harmonic Polynomials
89(3)
4.A.3 Associated Legendre Functions
92(1)
4.A.4 Generalized Laguerre Polynomials
93(1)
4.A.5 Surface Measure on the Sphere and Gamma Function
94(1)
4.A.6 Green Function of the Laplacian
95(1)
5 Regularization and Symmetry
96(13)
5.1 Moser Method
97(5)
5.2 Souriau Method
102(3)
5.2.1 Fock Parameters
104(1)
5.2.2 Bacry--Gyorgyi Parameters
105(1)
5.3 Kustaanheimo--Stiefel Transformation
105(4)
II Group--Geometric Theory
109(126)
6 Conformal Regularization
110(33)
6.1 The Conformal Group
111(4)
6.2 The Compactified Minkowski Space
115(4)
6.3 The Cotangent Bundle to Minkowski Space
119(10)
6.4 Regularization of the Kepler Problem
129(14)
7 Spinorial Regularization
143(18)
7.1 The Homomorphism SU(2, 2) → SO(2, 4)
143(7)
7.1.1 Two Bases for su(2, 2)
145(2)
7.1.2 SU(2, 2) and Compactified Minkowski Space
147(3)
7.2 Return to the Kustaanheimo--Stiefel Map
150(6)
7.3 Generalized Kustaanheimo--Stiefel Map
156(5)
8 Return to Separation of Variables
161(31)
8.1 Separable Orthogonal Systems
161(9)
8.1.1 Stackel Theorem
162(2)
8.1.2 Eisenhart Theorem
164(5)
8.1.3 Robertson Theorem
169(1)
8.2 Finding Coordinate Systems Separating Kepler Problem
170(7)
8.2.1 Spherical Coordinates
173(1)
8.2.2 Parabolic Coordinates
173(2)
8.2.3 Elliptic Coordinates
175(1)
8.2.4 Spheroconical Coordinates
176(1)
8.3 Integrable Perturbations
177(15)
8.3.1 Euler Problem
179(10)
8.3.2 Stark Problem
189(1)
Appendix
8.A Jacobian Elliptic Functions
190(2)
9 Geometric Quantization
192(19)
9.1 Multiplier Representations
194(3)
9.2 Quantization of Geodesies on the Sphere
197(8)
9.3 Quantization of the Kepler Problem
205(6)
10 Kepler Problem with Magnetic Monopole
211(24)
10.1 Nonnull Twistors and Magnetic Monopoles
212(13)
10.1.1 Bound Motions
219(3)
10.1.2 Unbound Motions
222(1)
10.1.3 Quantization
223(2)
10.2 The MICZ System
225(3)
10.3 The Taub-NUT System
228(4)
10.4 The BPST Instanton
232(3)
III Perturbation Theory
235(86)
11 General Perturbation Theory
236(32)
11.1 Formal Expansions
237(8)
11.1.1 Lie Series and Formal Canonical Transformations
237(5)
11.1.2 Homological Equation and its Formal Solution
242(3)
11.2 The Convergence Problem
245(23)
11.2.1 Convergence of Lie Series
247(2)
11.2.2 Homological Equation and its Solution
249(4)
11.2.3 Kolmogorov Theorem
253(9)
11.2.4 Nekhoroshev Theorem
262(2)
Appendices
11.A Results from Diophantine Theory
264(1)
11.B Cauchy Inequality
265(3)
12 Perturbations of the Kepler Problem
268(25)
12.1 A More Convenient Hamiltonian
270(6)
12.2 Normalization (or Averaging) Method
276(8)
12.3 Numerical Integration
284(9)
12.3.1 Symbolic Manipulation
285(3)
12.3.2 Compiling Equations
288(3)
Appendices
12.A Variation of the Constants
291(1)
12.B The Stabilization Method
291(2)
13 Perturbations with Axial Symmetry
293(28)
13.1 Reduction of Orbit Manifold
293(9)
13.2 Lunar Problem
302(9)
13.3 Stark and Quadratic Zeeman Effect
311(2)
13.4 Satellite around Oblate Primary
313(8)
IV Appendices
321(1)
A Differential Geometry
322(40)
A.1 Rudiments of Topology
322(2)
A.2 Differentiable Manifolds
324(7)
A.2.1 Definition
324(3)
A.2.2 Tangent and Cotangent Spaces
327(2)
A.2.3 Push--forward and Pull--back
329(2)
A.3 Tensors and Forms
331(10)
A.3.1 Tensors
331(1)
A.3.2 Forms and Exterior Derivatives
332(3)
A.3.3 Lie Derivative
335(2)
A.3.4 Integration of Differential Forms
337(4)
A.4 Distributions and Frobenius Theorem
341(3)
A.5 Riemannian, Symplectic and Poisson Manifolds
344(10)
A.5.1 Riemannian Manifolds
344(4)
A.5.2 Symplectic Manifolds
348(4)
A.5.3 Poisson Manifolds
352(2)
A.6 Fibre Bundles
354(8)
A.6.1 Definition
354(3)
A.6.2 Principal and Associated Fibre Bundles
357(5)
B Lie Groups and Lie Algebras
362(16)
B.1 Definition and Properties
362(4)
B.2 Adjoint and Coadjoint Representation
366(3)
B.3 Action of a Lie Group on a Manifold
369(3)
B.4 Classification of Lie Groups and Lie Algebras
372(3)
B.5 Connection on a Principal Bundle
375(3)
C Lagrangian Dynamics
378(10)
C.1 Lagrange Equations
378(4)
C.2 Hamilton Principle
382(1)
C.3 Noether Theorem
383(1)
C.4 Reduced Lagrangian and Maupertuis Principle
384(4)
D Hamiltonian Dynamics
388(35)
D.1 From Lagrange to Hamilton
388(2)
D.2 The Hamilton--Jacobi Integration Method
390(8)
D.2.1 Canonical Transformations
390(2)
D.2.2 Hamilton--Jacobi Equation
392(1)
D.2.3 Geometric Description
393(4)
D.2.4 The Time--dependent Case
397(1)
D.3 Symmetries and Reduction
398(10)
D.3.1 The Moment Map
399(3)
D.3.2 Reduction of Symplectic Manifolds
402(2)
D.3.3 Reduction of Poisson Manifolds
404(4)
D.4 Action--Angle Variables
408(15)
D.4.1 Arnold Theorem
409(7)
D.4.2 Degenerate Systems
416(1)
D.4.3 Monodromy
417(6)
Bibliography 423(10)
Index 433
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